English Course for Mathematicians

Ex 1. Replace the Object Clause with the Complex Object. Follow the models.

Model 1. I want to measure this line with a meter. (he) I want him to measure the line with a meter.

1.They want to complete this experiment as soon as possible. (she)

2.The supervisor requires to write this thesis in three years. (we)

3.The teacher expects to speak at the students’ conference. (they)

4.I want to assign the letter X to the vertical line. (you)

5.I should like to draw a circle. (he)

6.She means first to inscribe a square. (we)

7.They expect to extend the line in one direction. (I)

8.We desire to speak on the properties of this circle. (she)

9.They would like to inscribe a square in a circle. (we)

Model 2. We believe that the experiment is of great importance.

We believe the experiment to be of great importance.

1.The teacher knows that they intersect two-number lines at the zero point.

2.I know that they have solved the equation easily.

3.The professor watches that his students carry out the experiments carefully.

4.I expect that they describe the circle in more details.

5.He believes that she designates a circle by a special symbol.

6.We know that mathematics has become man’s second language.

7.They heard that the scientists had discussed a new theory.

Ex 2. In the sentences to follow look for the Complex Object and then translate them into Russian.

1.We know all points on the circle to be equidistant from the center point.

2.They can assume a line to be defined as an infinitely large set of points.

3.The teacher heard them discuss a new theorem.

4.We consider both theories to be necessary, though they are contradictory.

5.I expect this equation to have a different solution.

6.Some prominent scientists believe many problems of maths to be solved in the 21st century.

7.We want the students to learn and revise the rules regularly not only before the exams.

8.The professor desired his postgraduate student to apply a new method of investigation.

9.We watched the professor (him) draw a new axis in order to prove the theorem considered.

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10.They expected a solution to be found as soon as possible.

11.Mathematicians have found the ratio of the circumference to a diameter to be the same for all circles.

12.They didn’t expect trigonometric functions to be so complicated.

Complex Subject (именительный падеж с инфинитивом)

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5.Like terms are expected to be arranged in a similar way.

6.A proper solution of this equation is likely to be obtained.

7.The students appeared to be unable to carry out these complex calculations.

8.The line drawn perpendicular to a radius through an endpoint of the radius is known to be a tanget of the circle.

9.Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are said to be equal in length.

10.A circle is known to be a set of points in a plane each of which is equidistant from some given point called the center.

11.Every point at a distance less than r from 0 is said to be inside the circle.

12.The sets are supposed to be designated by capital letters A.B.C..

13.Two circles which have two common points are said to intersect each other.

Ex. 4. Change according to the model:

It is believed that he is a hard-working student.

He is believed to be a hard-working student.

1.It is expected that all second-year students will pass the exams successfully.

2.It is known that the experiments have been finished in time.

3.It is certain that this problem will be solved very soon.

4.It is likely that he has given them wrong instructions.

5.It is reported that the postgraduates of our Faculty have finished their investigations.

6.It seems that she is a very talented researcher.

7.It happened so that the error was easily detected.

8.It is expected that the delegation of prominent foreign mathematicians will arrive at our University next week.

Pre-Reading Activity

Guess the meaning of the following words:

coordinate [kou′O:dnIt] (n), diameter [daI′xmItq] (n), arc [a:k] (n), perpendicular [‚pq:pqn′dikjulq] (n).

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Read and learn the basic vocabulary terms

circle [sWkl] n – круг

curved line [kWvd] – кривая линия

radius [reIdIqs] radii [′reIdIaI] pl., n – радиус, радиусы half-line [hRflaIn] n – полупрямая, луч

measure [meZq] v – мерить, измерять reference-line n – базисная линия equidistant adj – равноудаленный chord [kLd] n – хорда

secant [sJkqnt] n секущая secant line

come form v – происходить, иметь происхождение inscribe v – вписывать

bisect v – делить пополам arc [Rk] – дуга

tangent [txnGqnt] line – касательная прямая annulus [′xnjulqs] n – (плоское) круговое кольцо

ratio [reISIou] n – соотношение, коэффициент, отношение

Memorize the following word-combinations

a major arc – большая из двух дуг (окружности) a minor arc – меньшая из двух дуг

internally tangent circles – круги, касающиеся внутренним образом externally tangent circles – круги, касающиеся внешним образом a circumscribed polygon – описанный многоугольник

an inscribed polygon – вписанный многоугольник a regular octagon – правильный восьмиугольник

to arrive at a more precise definition – (для того) чтобы прийти к более точному определению

are subtended – стянуты

by a similar process – подобным образом in other words – иными словами

as closely as desired – так близко, как хотелось бы

no matter how short an arc is – какой бы короткой ни была дуга

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Reading Activity

The Circle and Regular Polygons

Let’s turn now to the simplest of all curved lines, the circle. We shall study its properties and its relation to straight lines and to figures made up of straight lines, especially polygons.

In a plane all the points at a given distance from a given fixed point are said to form a circle. A circle is a set of points in a plane, all of which are the same distance from a given point.

The fixed point Ο is called the CENTER of the circle, from which all other points are equidistant. The distance r is called the RADIUS. A radius is a line segment from the center of a circle to a point on the circle.

Every point at a distance r from О is said to be on the circle. Every point at a distance less than r from Ο is said to be inside the circle, and every point at a distance greater than r from Ο is said to be outside the circle. If the center of the circle is taken as the origin of a rectangular network, it follows from the Pythagorean Theorem that the coordinates (x, y) of every point P of the circle will satisfy the equation x2 + y2 = r2. This equation is the equation of the circle.

On any half-line with end-point Ο there is a point at the distance r from Ο. We may select one such half-line – for example, OX in Fig.1 as a reference line from which to measure the angles to all other such half-lines. If we measure these angles in degrees, then on every half-line which makes an angle

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arc that corresponds to the greater central angle. Except for the end-points P and Q, all the points of the first arc PQ are sometimes called the minor arc which are distinct from the points of the second arc PQ which are sometimes called the major arc. If the two central angles POQ are equal, each of the two corresponding area PQ is called a semicircle.

Instead of speaking of the perimeter of a circle, we usually use the term circumference to mean the distance around a circle. We cannot find the circumference of a circle by adding the measure of the segments, because a circle does not contain any segments. No matter how short an arc is, it is curved at least slightly. Mathematicians have discovered, that the ratio of the circumference (c) to a diameter (d) is

the same for all circles and is

diameter is equal to twice the length of a radius), which is the same for all circles, is designated by π.

By using the multiplication property of equation, we obtain the following:

c = πd or c = 2πr.

Fig. 3

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Post-Reading Activity

Ex. 5. Answer the following questions.

1.How can we define a circle?

2.What is the center of a circle?

3.What is a radius?

4.When do we say that a point is on the circle, inside the circle and outside the circle?

5.How can you formulate the equation of the circle using the Pythagorean Theorem?

6.What is an arc?

7.When is an angle called a central angle?

8.What kind of line segment is a chord?

9.What is the longest chord in a circle? Give the definition of this figure.

10.What is a tangent to a circle?

11.What kind of chord is a secant? Give the definition of a secant.

12.What is a circumference? Give the formula of a circumference.

Ex. 6. Match the English words and words combinations with their Russian equivalents.

Ex. 7. Mark the following as True or False. Use the introductory phrases:

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1.In a plane all the points at a given distance from a given fixed point are said to form a chord.

2.The diameter passing through the circle centre, is the shortest chord in a circle.

3.Angle POQ is called a central angle because its vertex is at the centre of the circle.

4.A tangent to a circle is a curved line that touches the circle at two points.

5.A secant is an extended chord: a straight line cutting the circle at a single point.

6.Mathematicians have discovered that the ratio of the circumference to a diameter is the same for all circles.

7.The circumference of a circle may be defined as the limit of the perimeter of an circumscribed regular polygon.

8.The equation x2 + y2 = r2 is the equation of the semicircle.

Ex. 8. Fill in the blanks with necessary words and word combinations. Mind there are two extra words.

1.A circle is ... in a plane, all of which are the same distance from a ... point.

2.A radius is ... from the center of a circle to a point ... .

3.Tangent circles are two circles that ... ... at one point.

4.A ... circle is a circle passing ... each ... of a polygon.

5. ... all the points of a given distance from a given ... point ... to form a circle.

6.The word ... comes from a Latin word ... “bow”.

7.A diameter is ... which ... the center to any points on the circle.

8.We cannot find ... of a circle by adding ... of the segments.

Ex. 9. Match the definitions of the circles with their names.

Based upon their relative positions, two circles in a plane or a circle and a polygon have special names.

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Ex. 10. Let us revise the Degrees of Comparison. Give the best English equivalents for the words in parentheses.

1.A circle is (самая простая) of all curved lines.

2.Every point at a distance (больше) than radius (говорят) to be outside the circle.

3.A secant segment is a line segment with an endpoint in the exterior of a circle, and the other endpoint on the circle, (самой далекой) from the external point.

4.Tom comes top in all the exams – he must be (самый умный) student in the group.

5.(Чем меньше) students think, (тем больше) they talk.

6.How are you today? – I’m very (хорошо), thanks.

7.Is this proof (более правильно)?

8.Peter speaks English (наиболее бегло) of all the students in this group.

9.(Чем больше) I learn, (тем больше) I forget and (тем меньше) I know.

10.(Чем скорее) the problem is solved, (тем лучше).

11.This contribution of the ancient Greeks is (намного больше, чем) the formulas of the Egyptians.

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